Frequently Asked Question

Residuals are the primary diagnostic tool for judging whether a steady-state simulation has converged — but they're often misread, so it's worth understanding what they actually represent.

What a residual is: At each iteration, the solver checks how well the current solution satisfies each governing equation across all cells. The residual is a normalized measure of that imbalance — essentially, "how far off is this equation from being perfectly satisfied right now." As iterations proceed and the solution improves, residuals should trend downward.

Typical convergence targets:

• Many general-purpose problems are considered reasonably converged when residuals drop by 3 orders of magnitude (e.g., from 1e-1 down to 1e-4)
• More precision-sensitive applications (detailed heat transfer, acoustics, small force differences) often require tighter targets, sometimes 1e-6 or lower
• What counts as "converged enough" ultimately depends on what decision the results will inform — a rough comparative study needs less precision than a final design validation

Why residuals alone aren't sufficient:

• A residual can appear to plateau at a stable but non-negligible value without truly being converged, especially in cases with inherent unsteadiness being forced into a steady-state solve
• Residuals are a numerical measure of equation balance, not directly a measure of whether your engineering quantity of interest (lift, drag, pressure drop, outlet temperature) has stabilized

Better practice — combine residuals with monitor points: Track a specific physical output relevant to your study (force coefficient, mass-averaged outlet temperature, pressure drop across the domain) alongside residuals. True convergence shows both: residuals flattened at an acceptably low level, AND your monitored quantity stable within an acceptable tolerance over many successive iterations (not still trending up or down).

Red flags suggesting the "solution" isn't real:

• Residuals plateau but oscillate in a stable repeating pattern (often indicates real unsteady physics — see Article 2)
• Monitored output values continue slowly drifting even though residuals look flat
• Mass or energy imbalance across the domain remains non-trivial despite low residuals

Numerical error and convergence

A numerical solution approximates the continuous conservation laws on a finite set of cells or control volumes. Discretization error depends on characteristic size, stretching, skewness, alignment, interpolation, boundary treatment, and the output being measured. Iterative convergence is different from mesh convergence: a solver can reduce algebraic residuals while the engineering quantity continues to drift.

φ is the reported quantity and h is a characteristic mesh size. Use an analytical or benchmark reference when possible; otherwise report a systematic refinement study.

Worked example

A pressure drop is 104 Pa on a coarse grid, 101 Pa on a medium grid, and 100 Pa on a fine grid. The medium-to-fine change is approximately 1%, while the coarse-to-medium change is approximately 3%. Report mesh sizes, refinement ratio, residual state, and whether the 1% change meets the design tolerance.

Check: refine where gradients or the engineering output are sensitive, not only where the global cell count is convenient.